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Computer modelling of pentoxifylline release from a cylinder-shaped hydroxylapatite carrier

Computer modelling of pentoxifylline release from a cylinder-shaped hydroxylapatite carrier The main objective of this article is to present the basic stages of creating a computer simulation of medication release from an undegraded carrier. Additionally, an innovative approach was presented to the construction of a porous material used for making the carrier for the medication. The programme presented here is based on Cellular Automata, used as a tool for modelling physical, chemical and biological phenomena which change in time. The results of in-silico calculations have been compared with empirical results. KEYWORDS: modeling, Cellular Automata, porous material, solubility. 1. Introduction The process of drug release from undegraded carriers is a very complex issue, influenced by a large number of diverse factors. This problem is of increasing significance in human life, since the type of drug is equally important as the form of drug administration. It must also be noted that the selection of the optimum dose and precise delivery of drug to the right location within an organism is of crucial importance to a successful therapy. In the case of drugs administered in the form of tablets, the initial level of active substance in human organism is quickly released, reaching a high concentration (Figure 1a). After reaching the peak level, the concentration will fall exponentially until another dose is administered. This standard method of drug delivery has a number of drawbacks. Firstly, during the initial phase the drug concentration may become too high, sometimes reaching harmful levels. In many cases, zero-order release kinetics is desired, assuring constant concentration of a substance in a given period. Secondly, the drug should be delivered to a key location within the organism for therapeutic effect, which cannot be ensured with the use of traditional drug administration methods [1]. The drug delivery on relevant carriers offers a number of advantages which cannot be achieved by using standard methods. First of all, the drug is released directly where it is needed to act. The risk of active substance overdose or of side effects is greatly reduced. Moreover, sophisticated drug administration methods provide the possibility of taking informed decisions on the quantities of drugs released into an organism for a pre-determined period, even as long as several weeks [3]. Figure 1. Drug release profile: a) tablet form b) drug delivery by controlled release method [2]. Other advantages offered by this method include the maintenance of active substance levels within optimum range (Figure 1b), as well as appropriate utilisation of the drug [4,5]. The studies [6,7,8,9,10,11] describe the application of ceramic and polymer biomaterials as drug carriers in the bone, brain, breast, ovarian and prostate cancer therapy. The applications of controlled drug release referred to above confirm that the method is developing dynamically and becoming more and more important in the therapeutic process. It must be noted, however, that to acquire the license for medical use of any new agent it is necessary to perform numerous tests and to meet rigorous requirements. In studies on new materials to be used as carriers, it is necessary to examine the bio-compliance of such materials, i.e. whether it has cytotoxic, carcinogenic, allergenic or immunogenic action. Additionally, it is also necessary to test the material in biological environment, i.e. to examine its degradation products. The study was conducted with the use of a hydroxylapatite, a ceramic material, the bio-compatibility and bio-compliance of which have been confirmed by lab tests [12,13]. 2. Formulation of the problem Nowadays, attempts are frequently made to introduce electronic devices into diverse fields of human life. Therefore, it seems of no surprise that computers also found their way to medical science [26]. The increasing number of studies are also undertaken to apply computer simulations for anticipating the profiles of drug release from carriers [14,15,16]. The most frequently used methods include that of Monte Carlo and Cellular Automata. The most frequently analysed carriers are micro-spheres made of polymeric materials which undergo degradation within a living organism. In such cases, the drug is distributed evenly within a micro-sphere (homogenous system). However, to date no simulations have been performed using modelled carriers made of hydroxylapatite in the form of a heterogeneous system (with the drug present in a separate phase). The objective of this article is to present the model created and the simulation of controlled release of pentoxifylline from a hydroxylapatite carrier in the form of a heterogeneous system, as well as the comparison of study results with lab data. 3. Cellular Automata The father of Cellular Automata is Janos von Neumann, who developed the concept of self-replicating model in the 1940s. Numerous scientists, such as Stanislaw Ulam, Edgar Frank Codd and Stephan Wolfram displayed interest in a given mathematic modelling tool. At present, Cellular Automata are applied in modelling physical, chemical and biological phenomena which take place with the participation of numerous interacting systems and change with time [17,27]. Cellular Automata are applied in numerous fields of science, such as i.e. chemistry (synthesis of chemical compounds), geology (the so-called selforganising critical phenomenon), biomedical engineering (bone regrowth simulations), as well as sociology (impact of parental care). Cellular Automaton is a dynamic mathematical model, wherein time, space and states represent discrete values. In each step, the Cellular Automaton is changing the state of its cells. This step is referred to as the system evolution. Each cell is assigned a state from a finite set of states. In order for a Cellular Automaton to correctly reflect the phenomenon that it simulates, it is necessary to: identify the initial states of all cells at time t = 0; determine a set of rules according to which the automaton may evolve. 4. Description of the Cellular Automaton used in the application The Cellular Automaton used in the application is a two-dimensional, singlelayer automaton with a square grid. Permissible cell states are presented in Table 1. 4.1. Diffusion model ­ Brownian motion Particles suspended within the medium move around at random. Such movement is referred to as Brownian motion [19]. The most important properties of this phenomenon include [19]: the velocity of motion which is inversely proportional to the particles' size, the medium which plays an important role in the motion of molecules, in particular its viscosity, stability in time, the motion is maintained for as long as the molecules remain within the medium, independence from the influence of external forces (such as i.e. electric or gravitational fields). Despite the fact that Brownian motion is actually a highly complicated process, it is frequently examined with the use of simple models. One of such models is called "drunk random walker". In each iteration, the displacement of a molecule is the same (regardless of the direction), and after each step a new direction is selected (all directions being equal). As it turns out, the distribution of molecules in space in this model corresponds to the Gauss distribution. Calculations also indicated that the average displacement at time t is proportional tot. The relation referred to above is met regardless of whether the dislocation in each iteration is the same or not (dislocation value may be selected at random in keeping with Gauss distribution). Table 1. The table presents a five-element set of states of Cellular Automaton Name of state Buffer (solvent) Graphic representation Description of state Cell representing a solvent. Medium within which particles of medication are diffused. A cell in hydroxylapatite state represents the material of which the block was made. The state does not change throughout the entire simulation. The cell presents free spaces within a block. In the course of solvent inflow, the state may be changed to buffer. Cell representing an undissolved drug. As a result of contact with cells in buffer state, the drug will change its state to dissolved. Cell representing molecules of dissolved drug. It may change its location within buffer-filled pores (diffuse). Upon leaving the block, its state is changed to buffer. Material (hydroxylapatite) Pore Solid-state drug Dissolved drug The study [16] applied the movement pattern of drug molecules based on Brownian motion. The authors focused on the problem of simulating controlled release of drugs from a biodegradable polymeric micro-sphere. The article offered a model of three-dimensional Cellular Automaton using von Neumann's neighbourhood. The possible states of cells are: polymer, solvent, pore, solid-state drug and dissolved drug. In the course of a single iteration, a cell in the state of a dissolved drug makes 400 movements by changing positions with a random neighbour in solvent state (the so-called Brownian motion). Upon leaving the boundaries of a sphere with the diameter of (d being the dimension of cell automaton grid), the cell will change its state to solvent. The Brownian motion-based algorithm of molecules' diffusion described above was applied in a modified form by the authors in the model of drug release from a hydroxylapatite carrier. In order to provide the best possible representation of actual behaviour of molecules, the type of neighbourhood was changed in the solvent. In the case of a two-dimensional automaton, von Neumann's neighbourhood allows movement in only 4 directions, whereas in reality, a molecule may move in any direction. For this reason, the best solution is to use Moore's neighbourhood. To avoid the problem of conflicts, asynchronous automaton was used. Additionally, the so-called "naive" method was applied, differing from Brown's method by reducing the number of location changes from 400 to 1. Further on, the automaton is asynchronous and operates based on Moore's neighbourhood. 4.2 Drug dissolution model The process of dissolution, although fairly complicated in physico-chemical terms, is usually presented in a simple way in simulations. In the study referred to above [16], authors modelled the drug release from a polymer micro-sphere using Cellular Automata. The rule of transition describing the process of drug dissolution is very simple: a cell in the state of a solid drug will change state to a dissolved drug when at least one of its neighbours is in solvent state. Other studies presented dissolution as similar models [13]. References also quotes cases when no cells in solid drug state are present in the model. In such cases, cells representing the drug are immediately ready for diffusion [12]. In the application, each cell in solid drug state was assigned the parameter of "solubility", determining the life span of such cell. In each iteration, where at least one of the neighbours was in buffer state, this parameter was decremented. As soon as the value of this parameter reaches zero, the state of the cell will change to dissolved drug. Hence, the lower the value of this parameter, the faster will the drug dissolve. However, in real life some areas of the drug may dissolve faster than the other. To take this situation into consideration, it was decided to introduce one more possibility. Drug solubility may be determined by normal distribution N(,). Hence, by setting the average value of the parameter of solubility , it is also possible to provide the standard deviation . Figure 2 presents sample distributions of the parameter of solubility. By adopting the proper standard deviation, it is possible to determine the range from which the "life span" of a cell will be selected. Figure 2. Distribution of the parameter of solubility compliant with normal distribution: (A) N(100.5), (B) N(300.30). 4.3 CA's models of water flow In the application described by the authors, the process of filling a block with buffer is diminutively short compared to the process of the drug dissolution and diffusion. For this reason, the authors did not focus on the description of that process. They only noted that FHP model had been used, representing in macroscopic scale the Navier-Stokes isotropic equations within the range of low-value Mach numbers [1]. 4.4 The algorithm of creating open porosity Porosity is a very important factor, influencing the kinetics of drug release. The dimensions and shape of pores notwithstanding, the process of drug release is also controlled by the volume of a block, i.e. its porosity index. The algorithm is required to create a system of open pores, allowing a slow drug release into the surrounding environment. The literature failed to provide a single, adequate model for the creation of a porous centre ensuring open porosity, comparably equal to total porosity. In this study it is of utmost significance, since in reality the parameter referred to above bears an important impact upon the kinetics of drug release. Pursuant to the above, the authors provided their own algorithm for creating porosity. The block is divided into 4 parts (quarters). To assure uniform distribution of pores throughout the entire centre, total open porosity must also be divisible by 4. The directions specified in algorithm description are the directions from Moore neighbourhood. The algorithm is based on effecting the following steps for each of the four parts of the block: 1. Select a random point situated at the edge of the circle which is a graphic representation of block edges. 2. Select at random the direction of channels depending upon the block quarter. The width of such channels will become the parameter provided by the user. Random selections are made with diverse probabilities, since the channels must "reach" the centre of the block containing the drug in solid state. 3. Select at random whether the pore is to be drawn in the shape provided. The size of the pore is selected at random from within the size range provided. 4. Repeat sections (2) and (3) until porosity does not come into contact with the drug in solid state. 5. Repeat section (1) followed by section (4) until the pre-determined porosity of the material is reached. Using the algorithm referred to above, a highly diversified micro-structure of material was reached. Figure 3 presents sample porosities of the medium. A) B) C) Figure 3. Sample medium microstructures achieved: A) thin channels with squareshaped pores, B) thin channels without pores, C) thin channels with circular pores. 5. Empirical laboratory data The results achieved are a part of a study on the application of the so-called Cracow hydroxylapatite bioceramics for manufacturing drug carriers, implanted into locations free of mechanical loads (due to low mechanic resistance of hydroxylapatite). In the course of tests carried out using both in vitro and in vivo methods, the kinetics of drug release from different blocks were analysed. The study also focused on the impact of manufacturing methods upon the properties of materials. Each experiment was carried out using identical, cylinder-shaped carriers, differing in terms of porosity and the shape and dimensions of pores. The blocks were made by single-axis pressing and baking of a mixture of hydroxylapatite and modifying agents. The addition of modifying agents to the power was made to provide appropriate porosity in the course of baking. The first of these modifying agents was Mg3(PO4)2*8H2O, and the other was Ca(PO3)2. The specification of block shapes is presented in Table 2. Table 2. Types of materials used to produce blocks In each block, a cylindrical channel was drilled, subsequently filled with appropriate medication. Finally, each orifice was sealed with bone wax, leaving the pores as the only exit route for the medication. All blocks had the same dimensions: cylinder diameter - 10 mm, cylinder height - 9 mm, diameter of drilled channel - 4 mm, length of drilled channel - 6 mm, active surface - 2.74±0.23 cm2. The study focused on the release of one medication ­ pentoxifylline (PTX). Pentoxifylline is an anticoagulant, improving blood circulation in tissues, applied in peripheral blood circulation disorders, mild ischemia of central nervous system, post-stroke conditions, diabetes-related retinal ischemia [3]. The channel drilled inside the block was filled with 50 mg of the selected medication and sealed with wax. Next, the sample was placed in a flask filled with 50 ml of phosphate buffer (pH = 7.4) immersed in a water bath (temperature of 37oC ). In pre-determined intervals of time, 3 ml samples of the solution were taken from the flask, the contents of which were supplemented after each sampling. The samples were used to determine the concentration of released medication using spectrophotometry or microdialysis. The experiment was repeated several times for the pentoxifylline-block set, and the results were averaged. Figure 4. The course of pentoxifylline release from 10FM-type blocks. 5. Simulation results The application implemented two diffusion methods based on Brownian motion. The methods differed in the number of changes in the location of diffusing molecule in the course of a single iteration. To start the analysis of results, it is necessary to define first the notion of the drug release rate. By definition, the rate of drug release is the change of drug quantity in time. However, the calculation of the first derivative of data representing the release profile will indicate that this rate is not constant. In most cases, the rate of release peaks in the first stage of the process. This section of the graph may be represented by a straight line. In further stages of the drug release process, saturation occurs and the rate of release will gradually fall. In this study, the drug release rate was adopted roughly as the angle of inclination of the profile in the point where 50 per cent of the drug has been released. A number of simulations was carried out to determine the impact of changing number of molecule dislocations upon the rate of the method. Figure 5 presents changes in the inclination angle of the curve for diffusion by Brownian motion method, differing in the number of dislocations. In each case, the value of the parameter of solubility was 1. Figure 5. The rate of drug release as a function of the number of molecule location changes per iteration in the course of diffusion by Brownian motion method. The visualisation of simulation data on the graph (Figure 5 indicates the existence of a certain relation. As it turns out, the results match very well the following function: +c 6.1 Matlab software was used to determine curve coefficients: 6.2 The results concerning changes in the drug release rate as a function of drug solubility assumed are presented below. The objective of the analysis was to confirm that diffusion models implemented in the study meet the expectations. It is obvious that increasing drug solubility shall accelerate the rate of release. It must be noted here that drug solubility is defined as the quantity of substance, expressed in milligrams, which can be dissolved in 100 millilitres of water. However, solubility as an application parameter is actually the reciprocal of drug solubility and means the number of iterations in the course of which a cell in solid drug state must have at least one neighbour in buffer state to transform into a cell in dissolved drug state. Hence, in the simulation, increasing drug solubility actually results in reducing the parameter of solubility. Figure 6 presents the release profiles for individual diffusion methods (5 repetitions), when the value of solubility parameter is 1. The specification also includes the Brownian motion method II referred to above. According to the method, the number of dislocations of drug molecules per iteration is 50. Figure 6. Release profiles for the value of solubility parameter of 1 (diffusion models based on Brownian motion method). A number of simulations was carried out for diffusion models featuring different values of solubility parameter. Each simulation was repeated several times to assure credible results. Next, the angle of inclination of the curve was used as a measure of the drug release rate, presenting the quantity of the drug released as a function of iteration number. The results for Brownian method I are presented in Figure 7. Figure 7. Diffusion by Brownian motion method. Release profiles for different values of the parameter of solubility. The detailed analysis of resultant graphs inclined the authors to reconsider the relation between the parameter of solubility and the drug release rate. Each diffusion method was matched with an exponential function: 6.3 The divergence indices calculated for each case indicated a very good match each time. Table 3. Relation between the parameter of solubility and the drug release rate. Matched exponents with divergence index calculated Figure 8. Diffusion by Brownian motion method. Relation between the parameter of solubility and the inclination angle of the release curve. In the study [12], the relation between porosity and the maximum rate of release is linear. The impact of porosity upon simulation results was analysed. The examination of results indicated that in the case of diffusion models applied, the relation between the inclination angle of the curve and the parameter of solubility is close to linear. To check whether this relation is accidental or actually true, linear regression method was applied. Next, the determination coefficient was calculated as a measure of match between the straight line and the data set. The coefficient falls within the range of 0-1 (or 0-100%). The closer the coefficient value to 1, the better the match. The second parameter used as a measure of match quality was the divergence index, described in Section 8 - Calibration. Also in this case, possible values fall within the range of 0-1, although in this particular case, the best possible value is 0. For Brownian motion methods (I and II) and for the naive method, over ten simulations have been carried out, for different values of solubility parameter and porosity. Next, linear regression method was applied to match the straight lines to data sets. The results for Brownian method I are presented in Figure 9. Figure 9. Relation between drug release rate and block porosity. Brownian motion method. Individual curves correspond to different values of the parameter of solubility. 7. Calibration of simulation results and empirical results The measure of success of a simulation is the best possible match of results achieved with experimental results. Hence, a criterion function must be introduced, making it possible to compare the results of several simulations. In [16], the so-called divergence index was proposed (Ic): where: E2 - experimental result for time t, St - simulation result for time t, t - time, N - number of points. The index in question is a number falling within the range from 0 to 1. The lower the index value, the better the match between the simulation and experimental results. In [16] it is suggested that Ic < 0.2 means a good match. A notable advantage offered by the divergence index is the possibility of comparing the results with a large number of data. For this reason, this coefficient was selected as one of the measures of match quality of two curves. The purpose of the calibration process is to find the relation between the number of simulation iterations and real time. In this case, sufficient experimental data is necessary. The data is used to verify whether the calibration was correct. The basic calibration-related problem is the small range of inclination angles of release curves for individual diffusion methods. Actually, the profiles may considerably differ depending upon drug solubility. In the case of pentoxifylline, the inclination angle of the curve is approx. 75° (Figure 10), whereas in the case of Brownian motion-based method, the angle falls within the range of from 41° to 0.8°. Figure 10. Relation between the quantity of released drug and time for pentoxifylline. Hence, the diffusion simulation methods offered (Brown I, II and naive) are incapable of covering the entire space of possible results. Neither do the available references on the subject provide such solutions. Researchers usually focus on a specific carrier and drug [13,41,14,16]. Following their example, calibration may be effected for a specific case. Obviously, for each diffusion method the ratio: number of iterations ­ time will be different. Figure 11 presents the profile of pentoxifylline release from type E block together with the results of simulations for different calibration methods. Figure 11. Pentoxifylline release from type E blocks inclusive of calibrated simulation results for diffusion methods. Figure 12 presents release profiles for PTX for different carrier systems, including sample post-calibration simulation results. Divergence indices for individual pairs of curves are also provided. A good match can be seen here, since index values are considerably below 0.2 and fall within the range from <0.1040 to 0.0577>. The matching of simulation results to a broader range of angles required the application of a method yielding better results than the Brownian movement method, allowing curve inclination angles of less than 41° to be reached. Figure 6 presents the changes in angles for Brownian movement methods differing in the number of molecule dislocations per iteration. The resultant range of results was very broad (75° - 1°). Due to the lack of mathematical relationship between drug solubility and drug release rate, precise calibration was not possible. Further in the study, the simplest variant was adopted, according to which the release rate is proportional to drug solubility. Based on the above assumption, the relation was determined between drug solubility and rate of drug release from blocks, for which sufficient number of data was available. FW5, FW10 and E-type blocks were used. Figure 12. Laboratory data for PTX, including calibrated simulation results. Apart from simulation results, the values of divergence indices are also provided The subsequent course of the calibration process is presented in Figure 13. First, the type of block (FW5, FW10, E) and drug solubility R (mg/100ml) are selected for which the simulation will be performed. Next, the relations presented in Table 4 are used to determine the expected profile inclination angle V (item 1 in Figure 13). The relation between the number of molecule dislocations per iteration and the rate of release in the course of diffusion by Brownian motion method is determined earlier using formula 6.2. Using the calculated angle (item 2) we can calculate the value of X sought (item 3), i.e. the number of molecule dislocations per iteration. For these parameters the simulation is carried out. Table 4. Assumptions adopted in the course of calibration regarding the relation between drug solubility and drug release rate for individual blocks Figure 13. Calibration method. The curve presents the relation between the number of molecule dislocations per iteration according to Brownian motion method and the drug release rate. The straight line represents the relation between drug solubility and drug release rate 8. Conclusions The objective of this study was to develop and validate a model of controlled release of pentoxifylline from cylinder-shaped hydroxylapatite carrier using Cellular Automata. In the course of the study, the analysis was performed of most important parameters of both the blocks and the drug, as well as of various models of buffer inflow, diffusion and dissolution of molecules. The main conclusions formulated in the course of the study are as follows: a) The research into diffusion using Brownian motion method indicated that these methods are stable and foreseeable. The use of the same simulation parameters yielded highly similar drug release profiles. b) The analysis of relation between the parameter of solubility and the drug release profile inclination angle indicated that Brownian motion methods and the naive method yield results falling within a broad range of angles. c) For the diffusion methods presented, the relation between drug solubility and drug release rate is exponential. d) The study confirmed that, as in reality, the simulation deals with two types of drug release: diffusion- or dissolution-controlled. e) f) The impact of porosity upon the drug release rate was found to be in compliance with reality. The drug release rate is directly proportional to block porosity. When the solubility parameter remains in compliance with normal distribution N(,) release rate will slightly grow as compared to the case when no such distribution is applied. The effect referred to above grows with the value of standard deviation . In vitro and in vivo experiments have been extensively used for a long time in numerous fields of science. At present, an increasing number of studies refer to research conducted with the use of computers, i.e. in silico. Evidence suggests that the significance of this type of research will grow as this technology develops. In the future, perhaps, experiments performed in silico will be quoted on equal terms with experiments in vitro and in vivo. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bio-Algorithms and Med-Systems de Gruyter

Computer modelling of pentoxifylline release from a cylinder-shaped hydroxylapatite carrier

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Abstract

The main objective of this article is to present the basic stages of creating a computer simulation of medication release from an undegraded carrier. Additionally, an innovative approach was presented to the construction of a porous material used for making the carrier for the medication. The programme presented here is based on Cellular Automata, used as a tool for modelling physical, chemical and biological phenomena which change in time. The results of in-silico calculations have been compared with empirical results. KEYWORDS: modeling, Cellular Automata, porous material, solubility. 1. Introduction The process of drug release from undegraded carriers is a very complex issue, influenced by a large number of diverse factors. This problem is of increasing significance in human life, since the type of drug is equally important as the form of drug administration. It must also be noted that the selection of the optimum dose and precise delivery of drug to the right location within an organism is of crucial importance to a successful therapy. In the case of drugs administered in the form of tablets, the initial level of active substance in human organism is quickly released, reaching a high concentration (Figure 1a). After reaching the peak level, the concentration will fall exponentially until another dose is administered. This standard method of drug delivery has a number of drawbacks. Firstly, during the initial phase the drug concentration may become too high, sometimes reaching harmful levels. In many cases, zero-order release kinetics is desired, assuring constant concentration of a substance in a given period. Secondly, the drug should be delivered to a key location within the organism for therapeutic effect, which cannot be ensured with the use of traditional drug administration methods [1]. The drug delivery on relevant carriers offers a number of advantages which cannot be achieved by using standard methods. First of all, the drug is released directly where it is needed to act. The risk of active substance overdose or of side effects is greatly reduced. Moreover, sophisticated drug administration methods provide the possibility of taking informed decisions on the quantities of drugs released into an organism for a pre-determined period, even as long as several weeks [3]. Figure 1. Drug release profile: a) tablet form b) drug delivery by controlled release method [2]. Other advantages offered by this method include the maintenance of active substance levels within optimum range (Figure 1b), as well as appropriate utilisation of the drug [4,5]. The studies [6,7,8,9,10,11] describe the application of ceramic and polymer biomaterials as drug carriers in the bone, brain, breast, ovarian and prostate cancer therapy. The applications of controlled drug release referred to above confirm that the method is developing dynamically and becoming more and more important in the therapeutic process. It must be noted, however, that to acquire the license for medical use of any new agent it is necessary to perform numerous tests and to meet rigorous requirements. In studies on new materials to be used as carriers, it is necessary to examine the bio-compliance of such materials, i.e. whether it has cytotoxic, carcinogenic, allergenic or immunogenic action. Additionally, it is also necessary to test the material in biological environment, i.e. to examine its degradation products. The study was conducted with the use of a hydroxylapatite, a ceramic material, the bio-compatibility and bio-compliance of which have been confirmed by lab tests [12,13]. 2. Formulation of the problem Nowadays, attempts are frequently made to introduce electronic devices into diverse fields of human life. Therefore, it seems of no surprise that computers also found their way to medical science [26]. The increasing number of studies are also undertaken to apply computer simulations for anticipating the profiles of drug release from carriers [14,15,16]. The most frequently used methods include that of Monte Carlo and Cellular Automata. The most frequently analysed carriers are micro-spheres made of polymeric materials which undergo degradation within a living organism. In such cases, the drug is distributed evenly within a micro-sphere (homogenous system). However, to date no simulations have been performed using modelled carriers made of hydroxylapatite in the form of a heterogeneous system (with the drug present in a separate phase). The objective of this article is to present the model created and the simulation of controlled release of pentoxifylline from a hydroxylapatite carrier in the form of a heterogeneous system, as well as the comparison of study results with lab data. 3. Cellular Automata The father of Cellular Automata is Janos von Neumann, who developed the concept of self-replicating model in the 1940s. Numerous scientists, such as Stanislaw Ulam, Edgar Frank Codd and Stephan Wolfram displayed interest in a given mathematic modelling tool. At present, Cellular Automata are applied in modelling physical, chemical and biological phenomena which take place with the participation of numerous interacting systems and change with time [17,27]. Cellular Automata are applied in numerous fields of science, such as i.e. chemistry (synthesis of chemical compounds), geology (the so-called selforganising critical phenomenon), biomedical engineering (bone regrowth simulations), as well as sociology (impact of parental care). Cellular Automaton is a dynamic mathematical model, wherein time, space and states represent discrete values. In each step, the Cellular Automaton is changing the state of its cells. This step is referred to as the system evolution. Each cell is assigned a state from a finite set of states. In order for a Cellular Automaton to correctly reflect the phenomenon that it simulates, it is necessary to: identify the initial states of all cells at time t = 0; determine a set of rules according to which the automaton may evolve. 4. Description of the Cellular Automaton used in the application The Cellular Automaton used in the application is a two-dimensional, singlelayer automaton with a square grid. Permissible cell states are presented in Table 1. 4.1. Diffusion model ­ Brownian motion Particles suspended within the medium move around at random. Such movement is referred to as Brownian motion [19]. The most important properties of this phenomenon include [19]: the velocity of motion which is inversely proportional to the particles' size, the medium which plays an important role in the motion of molecules, in particular its viscosity, stability in time, the motion is maintained for as long as the molecules remain within the medium, independence from the influence of external forces (such as i.e. electric or gravitational fields). Despite the fact that Brownian motion is actually a highly complicated process, it is frequently examined with the use of simple models. One of such models is called "drunk random walker". In each iteration, the displacement of a molecule is the same (regardless of the direction), and after each step a new direction is selected (all directions being equal). As it turns out, the distribution of molecules in space in this model corresponds to the Gauss distribution. Calculations also indicated that the average displacement at time t is proportional tot. The relation referred to above is met regardless of whether the dislocation in each iteration is the same or not (dislocation value may be selected at random in keeping with Gauss distribution). Table 1. The table presents a five-element set of states of Cellular Automaton Name of state Buffer (solvent) Graphic representation Description of state Cell representing a solvent. Medium within which particles of medication are diffused. A cell in hydroxylapatite state represents the material of which the block was made. The state does not change throughout the entire simulation. The cell presents free spaces within a block. In the course of solvent inflow, the state may be changed to buffer. Cell representing an undissolved drug. As a result of contact with cells in buffer state, the drug will change its state to dissolved. Cell representing molecules of dissolved drug. It may change its location within buffer-filled pores (diffuse). Upon leaving the block, its state is changed to buffer. Material (hydroxylapatite) Pore Solid-state drug Dissolved drug The study [16] applied the movement pattern of drug molecules based on Brownian motion. The authors focused on the problem of simulating controlled release of drugs from a biodegradable polymeric micro-sphere. The article offered a model of three-dimensional Cellular Automaton using von Neumann's neighbourhood. The possible states of cells are: polymer, solvent, pore, solid-state drug and dissolved drug. In the course of a single iteration, a cell in the state of a dissolved drug makes 400 movements by changing positions with a random neighbour in solvent state (the so-called Brownian motion). Upon leaving the boundaries of a sphere with the diameter of (d being the dimension of cell automaton grid), the cell will change its state to solvent. The Brownian motion-based algorithm of molecules' diffusion described above was applied in a modified form by the authors in the model of drug release from a hydroxylapatite carrier. In order to provide the best possible representation of actual behaviour of molecules, the type of neighbourhood was changed in the solvent. In the case of a two-dimensional automaton, von Neumann's neighbourhood allows movement in only 4 directions, whereas in reality, a molecule may move in any direction. For this reason, the best solution is to use Moore's neighbourhood. To avoid the problem of conflicts, asynchronous automaton was used. Additionally, the so-called "naive" method was applied, differing from Brown's method by reducing the number of location changes from 400 to 1. Further on, the automaton is asynchronous and operates based on Moore's neighbourhood. 4.2 Drug dissolution model The process of dissolution, although fairly complicated in physico-chemical terms, is usually presented in a simple way in simulations. In the study referred to above [16], authors modelled the drug release from a polymer micro-sphere using Cellular Automata. The rule of transition describing the process of drug dissolution is very simple: a cell in the state of a solid drug will change state to a dissolved drug when at least one of its neighbours is in solvent state. Other studies presented dissolution as similar models [13]. References also quotes cases when no cells in solid drug state are present in the model. In such cases, cells representing the drug are immediately ready for diffusion [12]. In the application, each cell in solid drug state was assigned the parameter of "solubility", determining the life span of such cell. In each iteration, where at least one of the neighbours was in buffer state, this parameter was decremented. As soon as the value of this parameter reaches zero, the state of the cell will change to dissolved drug. Hence, the lower the value of this parameter, the faster will the drug dissolve. However, in real life some areas of the drug may dissolve faster than the other. To take this situation into consideration, it was decided to introduce one more possibility. Drug solubility may be determined by normal distribution N(,). Hence, by setting the average value of the parameter of solubility , it is also possible to provide the standard deviation . Figure 2 presents sample distributions of the parameter of solubility. By adopting the proper standard deviation, it is possible to determine the range from which the "life span" of a cell will be selected. Figure 2. Distribution of the parameter of solubility compliant with normal distribution: (A) N(100.5), (B) N(300.30). 4.3 CA's models of water flow In the application described by the authors, the process of filling a block with buffer is diminutively short compared to the process of the drug dissolution and diffusion. For this reason, the authors did not focus on the description of that process. They only noted that FHP model had been used, representing in macroscopic scale the Navier-Stokes isotropic equations within the range of low-value Mach numbers [1]. 4.4 The algorithm of creating open porosity Porosity is a very important factor, influencing the kinetics of drug release. The dimensions and shape of pores notwithstanding, the process of drug release is also controlled by the volume of a block, i.e. its porosity index. The algorithm is required to create a system of open pores, allowing a slow drug release into the surrounding environment. The literature failed to provide a single, adequate model for the creation of a porous centre ensuring open porosity, comparably equal to total porosity. In this study it is of utmost significance, since in reality the parameter referred to above bears an important impact upon the kinetics of drug release. Pursuant to the above, the authors provided their own algorithm for creating porosity. The block is divided into 4 parts (quarters). To assure uniform distribution of pores throughout the entire centre, total open porosity must also be divisible by 4. The directions specified in algorithm description are the directions from Moore neighbourhood. The algorithm is based on effecting the following steps for each of the four parts of the block: 1. Select a random point situated at the edge of the circle which is a graphic representation of block edges. 2. Select at random the direction of channels depending upon the block quarter. The width of such channels will become the parameter provided by the user. Random selections are made with diverse probabilities, since the channels must "reach" the centre of the block containing the drug in solid state. 3. Select at random whether the pore is to be drawn in the shape provided. The size of the pore is selected at random from within the size range provided. 4. Repeat sections (2) and (3) until porosity does not come into contact with the drug in solid state. 5. Repeat section (1) followed by section (4) until the pre-determined porosity of the material is reached. Using the algorithm referred to above, a highly diversified micro-structure of material was reached. Figure 3 presents sample porosities of the medium. A) B) C) Figure 3. Sample medium microstructures achieved: A) thin channels with squareshaped pores, B) thin channels without pores, C) thin channels with circular pores. 5. Empirical laboratory data The results achieved are a part of a study on the application of the so-called Cracow hydroxylapatite bioceramics for manufacturing drug carriers, implanted into locations free of mechanical loads (due to low mechanic resistance of hydroxylapatite). In the course of tests carried out using both in vitro and in vivo methods, the kinetics of drug release from different blocks were analysed. The study also focused on the impact of manufacturing methods upon the properties of materials. Each experiment was carried out using identical, cylinder-shaped carriers, differing in terms of porosity and the shape and dimensions of pores. The blocks were made by single-axis pressing and baking of a mixture of hydroxylapatite and modifying agents. The addition of modifying agents to the power was made to provide appropriate porosity in the course of baking. The first of these modifying agents was Mg3(PO4)2*8H2O, and the other was Ca(PO3)2. The specification of block shapes is presented in Table 2. Table 2. Types of materials used to produce blocks In each block, a cylindrical channel was drilled, subsequently filled with appropriate medication. Finally, each orifice was sealed with bone wax, leaving the pores as the only exit route for the medication. All blocks had the same dimensions: cylinder diameter - 10 mm, cylinder height - 9 mm, diameter of drilled channel - 4 mm, length of drilled channel - 6 mm, active surface - 2.74±0.23 cm2. The study focused on the release of one medication ­ pentoxifylline (PTX). Pentoxifylline is an anticoagulant, improving blood circulation in tissues, applied in peripheral blood circulation disorders, mild ischemia of central nervous system, post-stroke conditions, diabetes-related retinal ischemia [3]. The channel drilled inside the block was filled with 50 mg of the selected medication and sealed with wax. Next, the sample was placed in a flask filled with 50 ml of phosphate buffer (pH = 7.4) immersed in a water bath (temperature of 37oC ). In pre-determined intervals of time, 3 ml samples of the solution were taken from the flask, the contents of which were supplemented after each sampling. The samples were used to determine the concentration of released medication using spectrophotometry or microdialysis. The experiment was repeated several times for the pentoxifylline-block set, and the results were averaged. Figure 4. The course of pentoxifylline release from 10FM-type blocks. 5. Simulation results The application implemented two diffusion methods based on Brownian motion. The methods differed in the number of changes in the location of diffusing molecule in the course of a single iteration. To start the analysis of results, it is necessary to define first the notion of the drug release rate. By definition, the rate of drug release is the change of drug quantity in time. However, the calculation of the first derivative of data representing the release profile will indicate that this rate is not constant. In most cases, the rate of release peaks in the first stage of the process. This section of the graph may be represented by a straight line. In further stages of the drug release process, saturation occurs and the rate of release will gradually fall. In this study, the drug release rate was adopted roughly as the angle of inclination of the profile in the point where 50 per cent of the drug has been released. A number of simulations was carried out to determine the impact of changing number of molecule dislocations upon the rate of the method. Figure 5 presents changes in the inclination angle of the curve for diffusion by Brownian motion method, differing in the number of dislocations. In each case, the value of the parameter of solubility was 1. Figure 5. The rate of drug release as a function of the number of molecule location changes per iteration in the course of diffusion by Brownian motion method. The visualisation of simulation data on the graph (Figure 5 indicates the existence of a certain relation. As it turns out, the results match very well the following function: +c 6.1 Matlab software was used to determine curve coefficients: 6.2 The results concerning changes in the drug release rate as a function of drug solubility assumed are presented below. The objective of the analysis was to confirm that diffusion models implemented in the study meet the expectations. It is obvious that increasing drug solubility shall accelerate the rate of release. It must be noted here that drug solubility is defined as the quantity of substance, expressed in milligrams, which can be dissolved in 100 millilitres of water. However, solubility as an application parameter is actually the reciprocal of drug solubility and means the number of iterations in the course of which a cell in solid drug state must have at least one neighbour in buffer state to transform into a cell in dissolved drug state. Hence, in the simulation, increasing drug solubility actually results in reducing the parameter of solubility. Figure 6 presents the release profiles for individual diffusion methods (5 repetitions), when the value of solubility parameter is 1. The specification also includes the Brownian motion method II referred to above. According to the method, the number of dislocations of drug molecules per iteration is 50. Figure 6. Release profiles for the value of solubility parameter of 1 (diffusion models based on Brownian motion method). A number of simulations was carried out for diffusion models featuring different values of solubility parameter. Each simulation was repeated several times to assure credible results. Next, the angle of inclination of the curve was used as a measure of the drug release rate, presenting the quantity of the drug released as a function of iteration number. The results for Brownian method I are presented in Figure 7. Figure 7. Diffusion by Brownian motion method. Release profiles for different values of the parameter of solubility. The detailed analysis of resultant graphs inclined the authors to reconsider the relation between the parameter of solubility and the drug release rate. Each diffusion method was matched with an exponential function: 6.3 The divergence indices calculated for each case indicated a very good match each time. Table 3. Relation between the parameter of solubility and the drug release rate. Matched exponents with divergence index calculated Figure 8. Diffusion by Brownian motion method. Relation between the parameter of solubility and the inclination angle of the release curve. In the study [12], the relation between porosity and the maximum rate of release is linear. The impact of porosity upon simulation results was analysed. The examination of results indicated that in the case of diffusion models applied, the relation between the inclination angle of the curve and the parameter of solubility is close to linear. To check whether this relation is accidental or actually true, linear regression method was applied. Next, the determination coefficient was calculated as a measure of match between the straight line and the data set. The coefficient falls within the range of 0-1 (or 0-100%). The closer the coefficient value to 1, the better the match. The second parameter used as a measure of match quality was the divergence index, described in Section 8 - Calibration. Also in this case, possible values fall within the range of 0-1, although in this particular case, the best possible value is 0. For Brownian motion methods (I and II) and for the naive method, over ten simulations have been carried out, for different values of solubility parameter and porosity. Next, linear regression method was applied to match the straight lines to data sets. The results for Brownian method I are presented in Figure 9. Figure 9. Relation between drug release rate and block porosity. Brownian motion method. Individual curves correspond to different values of the parameter of solubility. 7. Calibration of simulation results and empirical results The measure of success of a simulation is the best possible match of results achieved with experimental results. Hence, a criterion function must be introduced, making it possible to compare the results of several simulations. In [16], the so-called divergence index was proposed (Ic): where: E2 - experimental result for time t, St - simulation result for time t, t - time, N - number of points. The index in question is a number falling within the range from 0 to 1. The lower the index value, the better the match between the simulation and experimental results. In [16] it is suggested that Ic < 0.2 means a good match. A notable advantage offered by the divergence index is the possibility of comparing the results with a large number of data. For this reason, this coefficient was selected as one of the measures of match quality of two curves. The purpose of the calibration process is to find the relation between the number of simulation iterations and real time. In this case, sufficient experimental data is necessary. The data is used to verify whether the calibration was correct. The basic calibration-related problem is the small range of inclination angles of release curves for individual diffusion methods. Actually, the profiles may considerably differ depending upon drug solubility. In the case of pentoxifylline, the inclination angle of the curve is approx. 75° (Figure 10), whereas in the case of Brownian motion-based method, the angle falls within the range of from 41° to 0.8°. Figure 10. Relation between the quantity of released drug and time for pentoxifylline. Hence, the diffusion simulation methods offered (Brown I, II and naive) are incapable of covering the entire space of possible results. Neither do the available references on the subject provide such solutions. Researchers usually focus on a specific carrier and drug [13,41,14,16]. Following their example, calibration may be effected for a specific case. Obviously, for each diffusion method the ratio: number of iterations ­ time will be different. Figure 11 presents the profile of pentoxifylline release from type E block together with the results of simulations for different calibration methods. Figure 11. Pentoxifylline release from type E blocks inclusive of calibrated simulation results for diffusion methods. Figure 12 presents release profiles for PTX for different carrier systems, including sample post-calibration simulation results. Divergence indices for individual pairs of curves are also provided. A good match can be seen here, since index values are considerably below 0.2 and fall within the range from <0.1040 to 0.0577>. The matching of simulation results to a broader range of angles required the application of a method yielding better results than the Brownian movement method, allowing curve inclination angles of less than 41° to be reached. Figure 6 presents the changes in angles for Brownian movement methods differing in the number of molecule dislocations per iteration. The resultant range of results was very broad (75° - 1°). Due to the lack of mathematical relationship between drug solubility and drug release rate, precise calibration was not possible. Further in the study, the simplest variant was adopted, according to which the release rate is proportional to drug solubility. Based on the above assumption, the relation was determined between drug solubility and rate of drug release from blocks, for which sufficient number of data was available. FW5, FW10 and E-type blocks were used. Figure 12. Laboratory data for PTX, including calibrated simulation results. Apart from simulation results, the values of divergence indices are also provided The subsequent course of the calibration process is presented in Figure 13. First, the type of block (FW5, FW10, E) and drug solubility R (mg/100ml) are selected for which the simulation will be performed. Next, the relations presented in Table 4 are used to determine the expected profile inclination angle V (item 1 in Figure 13). The relation between the number of molecule dislocations per iteration and the rate of release in the course of diffusion by Brownian motion method is determined earlier using formula 6.2. Using the calculated angle (item 2) we can calculate the value of X sought (item 3), i.e. the number of molecule dislocations per iteration. For these parameters the simulation is carried out. Table 4. Assumptions adopted in the course of calibration regarding the relation between drug solubility and drug release rate for individual blocks Figure 13. Calibration method. The curve presents the relation between the number of molecule dislocations per iteration according to Brownian motion method and the drug release rate. The straight line represents the relation between drug solubility and drug release rate 8. Conclusions The objective of this study was to develop and validate a model of controlled release of pentoxifylline from cylinder-shaped hydroxylapatite carrier using Cellular Automata. In the course of the study, the analysis was performed of most important parameters of both the blocks and the drug, as well as of various models of buffer inflow, diffusion and dissolution of molecules. The main conclusions formulated in the course of the study are as follows: a) The research into diffusion using Brownian motion method indicated that these methods are stable and foreseeable. The use of the same simulation parameters yielded highly similar drug release profiles. b) The analysis of relation between the parameter of solubility and the drug release profile inclination angle indicated that Brownian motion methods and the naive method yield results falling within a broad range of angles. c) For the diffusion methods presented, the relation between drug solubility and drug release rate is exponential. d) The study confirmed that, as in reality, the simulation deals with two types of drug release: diffusion- or dissolution-controlled. e) f) The impact of porosity upon the drug release rate was found to be in compliance with reality. The drug release rate is directly proportional to block porosity. When the solubility parameter remains in compliance with normal distribution N(,) release rate will slightly grow as compared to the case when no such distribution is applied. The effect referred to above grows with the value of standard deviation . In vitro and in vivo experiments have been extensively used for a long time in numerous fields of science. At present, an increasing number of studies refer to research conducted with the use of computers, i.e. in silico. Evidence suggests that the significance of this type of research will grow as this technology develops. In the future, perhaps, experiments performed in silico will be quoted on equal terms with experiments in vitro and in vivo.

Journal

Bio-Algorithms and Med-Systemsde Gruyter

Published: Jan 1, 2012

References